Weighting pattern

Represent a linear time-varying system's zero-state input–output action by a two-time kernel formed from output, state-transition, and input maps, reducing to an impulse-response convolution kernel in the time-invariant case.

Version

v2 · 2026-08-30 · History

Domain-specific #

3105

Origin domain

linear systems theory

Subdomain

state space input output representations

Core Idea

The weighting pattern of a continuous-time linear state-space system is the two-time kernel (T(t,sigma)=C(t)Phi(t,sigma)B(sigma)) that maps past input at time (sigma) to its zero-state contribution to output at time (t). Variation of constants propagates an input injected through (B(sigma)) from (sigma) to (t) by (Phi(t,sigma)), then observes it through (C(t)); integrating (T(t,sigma)u(sigma)) over the causal interval yields the forced output separately from (C(t)Phi(t,t_0)x(t_0))

Scope of Application

Weighting pattern applies when the analyst can specify a linear time-varying state-space system (dot x=A(t)x+B(t)u), (y=C(t)x), with state-transition matrix (Phi(t,sigma)) and establish that the two-time kernel equals the composed input, transition, and output maps and reproduces the linear system's zero-state input–output response under the declared continuous or discrete convention. The entry treats deterministic finite-dimensional linear systems under stated regularity assumptions. Stochastic kernels, nonlinear Volterra series, and optimization weights are separate objects.

Clarity

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because weighting pattern is older systems terminology and can be mistaken for statistical weights; the two-time causal kernel and realization role must be explicit. The disciplined statement is that the object counts as Weighting pattern exactly when the two-time kernel equals the composed input, transition, and output maps and reproduces the linear system's zero-state input–output response under the declared continuous or discrete convention

Manages Complexity

The abstraction compresses continuous and discrete kernels, scalar and multivariable systems, time-invariant impulse responses, direct-feedthrough extensions, finite-horizon operators, and multiple state-space realizations into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares continuous or discrete time, time variance, initial state, direct feedthrough, causality, input and output dimensions, realization order, stability, integrability, and equivalence and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning

1. Type the carrier. Establish a linear time-varying state-space system (dot x=A(t)x+B(t)u), (y=C(t)x), with state-transition matrix (Phi(t,sigma)) and reject examples from a different problem. 2. Lock the rule. Express that the two-time kernel equals the composed input, transition, and output maps and reproduces the linear system's zero-state input–output response under the declared continuous or discrete convention independently of one notation or implementation.

Knowledge Transfer

Transfer within linear systems theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from For zero initial state, (y(t)=int_{t_0}^{t}T(t,sigma)u(sigma),dsigma) with (T(t,sigma)=C(t)Phi(t,sigma)B(sigma)). to For a continuous-time LTI realization, (T(t,sigma)=Ce^{A(t-sigma)}B) for \(t\geqsigma\), so the zero-state response becomes convolution with the impulse response. demonstrates that continuity.

Relationships to Other Abstractions

Current abstraction Weighting pattern Domain-specific

Parents (1) — more general patterns this builds on

• Weighting pattern is a kind of Representation Prime

  The proposed strict upward parent is prime:representation.

Hierarchy path (1) — routes to 1 parentless root

• Weighting pattern → Representation → Abstraction

Neighborhood in Abstraction Space

Weighting pattern sits in a moderately populated region (59^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Feedback Control & Dynamical Systems (29 abstractions)

Nearest neighbors

• State-transition matrix — 0.90
• Linear dynamical system — 0.90
• Linear time-invariant system — 0.86
• State variable — 0.86
• Correlation integral — 0.86

Computed from structural-signature embeddings · 2026-09-08